Should I Worry About Error?
Every estimate has error
- Any $\bar{x}$ from a sample is only an estimate of the population mean $\mu$.
- It carries uncertainty — it won't equal $\mu$ exactly.
- Inference for means (Unit 7) mirrors inference for proportions (Unit 6).
- The tools are again intervals (estimate a range) and tests (judge a claim).
Variability isn't a mistake
- Sampling variability — the natural sample-to-sample wobble — is not an error.
- A mistake in data collection (a mis-typed value, a biased method) is different.
- Variability is expected and quantifiable; a mistake must be fixed, not measured.
- Inference accounts for variability, not for blunders.
Why we quantify it
- We use intervals and tests to put honest numbers on the uncertainty.
- A confidence interval says "the mean is plausibly in this range."
- A test says "this claimed mean is (or isn't) surprising."
- Both convert vague doubt into precise probability statements.
Enter the t-distribution
- For proportions we used $z$; for means we usually don't know $\sigma$.
- We estimate it with the sample SD $s_x$ — which adds extra uncertainty.
- That forces a new model: the $t$-distribution, a bell that's a bit wider than normal.
- The $t$ has heavier tails to account for estimating $\sigma$ from data.
We use $t$, not $z$, for means because $\sigma$ is unknown. Estimating it with $s_x$ adds uncertainty, so the $t$-distribution has slightly heavier tails than the normal — giving wider intervals and more cautious tests. Don't reach for $z$ just because the sampling distribution "looks normal"; a mean inference with unknown $\sigma$ is a $t$ procedure.
You measure the mean weight of $20$ apples: $\bar{x} = 150$ g, $s_x = 12$ g.
- You don't know the true population SD $\sigma$ — only the sample $s_x$.
- So inference about $\mu$ uses the $t$-distribution, not the normal $z$.
- The $t$ accounts for the extra uncertainty from estimating $\sigma$.
Every sample mean $\bar{x}$ estimates $\mu$ with uncertainty from sampling variability (not a data mistake). We quantify it with intervals and tests. Because $\sigma$ is usually unknown and estimated by $s_x$, mean inference uses the $t$-distribution — a bell with heavier tails than the normal.
The t-distribution: a wider bell
The t has heavier tails than the normal to reflect estimating σ.
Inference about a mean uses the t-distribution (not z) mainly because...
Estimating σ with s adds uncertainty → use t.
Sampling variability is a mistake in data collection that must be fixed.
Variability is natural and quantifiable — not a blunder.
Compared with the normal, the t-distribution has...
Heavier tails reflect the extra uncertainty from estimating σ.
Inference for means uses the same two tools as inference for proportions: intervals and tests.
Both quantify uncertainty via intervals and tests.
When σ is unknown, mean inference uses the ___-distribution (one letter).
The t-distribution handles unknown σ.